The physical meaning of momentum? That is somewhat of a philosophical question.
The reason we measure momentum is because there is an extremely useful principle associated with it: the conservation of momentum.

You can view momentum as velocity proportioned by mass, or force proportioned by time. Both are ways of viewing it.
If something is applying a force on you, it matters how long the force is applied. Sort of like it matters how long you have been falling.
If something is moving, it matters the mass of the moving body. If a bike and bus hit each other going equal speeds, it's not going to be a symmetric collision.
So basically momentum is really important when we are studying collisions and forces.

Momentum is the sum of the result of all previous forces acting on an object, and it reflects the fact that if those forces have set the object in motion, it will require another force or forces of similar magnitude and duration to stop it. It's a reflection of the tendency of things to keep doing what they've been doing.

Momentum is NOT "the force that keeps an object in motion"...
In very simple terms, momentum is how much "umpf" is behind a moving object. You know that mathematically momentum = mass * velocity....so think about what that means.
Let's assume we have two balls of the same size with masses of 1kg and 10kg respectively. If they are both moving with the same velocity which do you think will hurt more if it hits you? Clearly, the one with the higher mass because it has more momentum (more umpf!).
Now let's say the 1kg ball is moving at 10m/s and the 10kg ball is moving at 1m/s. Both have the same momentum so they will both hurt you about the same.
Now let's say the 1kg object is moving at 15m/s and the 10kg object is moving at 1m/s. Now the smaller 1kg object will hurt you more....because it now has more momentum than the 10kg ball.
I hope this helps :)